Determination of local changes of concentration of admixtures in fluid flow

ABSTRACT

The invention allows determining a local change of an admixture concentration in a fluid flow at an entrance to a measurement cell. The change of the admixture concentration in time inside the measurement cell is first determined for a fluid containing the admixture, the change of concentration of which in time at the entrance to the measurement cell is known. Then, an impulse response of the cell is found applying the deconvolution method. The change of the admixture concentration inside the measurement cell is then determined for a fluid being studied with an unknown concentration of the admixture at the entrance. The unknown concentration is determined using the impulse response of the measurement cell and the change of the admixture concentration inside the cell.

CROSS-REFERENCE TO RELATED APPLICATION

This application claims priority to Russian Application No. 2012137227 filed Sep. 3, 2012, which is incorporated herein by reference in its entirety.

FIELD

The invention is related to measurement methods and might be used in estimation of nonstationary concentration of admixtures in a particular stream point.

BACKGROUND

It is known that the presence of an admixture alters various properties of a carrying fluid, such as density, color, radioactivity, magnetic and thermal properties and electrical resistivity. So, a determination of physical properties allows estimating the concentration of an admixture; in particular, compositions of a saline solution, a water-oil emulsion and other mixtures may be determined by measuring the electrical resistivity. The following concentration determining techniques are applied: visual (by admixture color), indirect (by flow conductivity), etc., (see, for example, M. Levy, B. Berkowitz, Measurement and Analysis of Non-Fickian Dispersion in Heterogeneous Porous Media, Journal of Contaminant Hydrology//2003, 64, pp. 203-226; Gas B, Zuska J, Coufal P, van de Goor T. Optimization of the High-Frequency Contactless Conductivity Detector for Capillary Electrophoresis, Electrophoresis II 2002, v. 23, pp. 3520-3527).

The main problem of the known techniques is the averaged nature of changes, i.e., the considerable temporal interval (conditioned by the dimensions of the measurement cell) within which an admixture concentration may alter significantly. The suggested technique provides a higher admixture concentration determination precision without changing the configuration of a measurement cell.

The suggested technique for determining local changes of concentration of admixtures in fluid flow involves pumping a fluid through a measurement cell; the fluid contains an admixture the change of the concentration of which in time at an entrance to the measurement cell is known. The admixture concentration change in time is determined inside the measurement cell, then, an impulse response of the measurement cell is recovered applying the method of deconvolution. A fluid being studied is pumped through the measurement cell and a change of the admixture concentration in time in the fluid flow is determined inside the measurement cell. The change of the admixture concentration in time in the flow of the fluid being studied at the entrance to the cell is found with the following equation:

∫₀^(t)K(t − τ)I(τ) τ = R_(σ)(t)

where τ—an integration variable, t—time, I(t)—change of the admixture concentration in the flow of the fluid being studied at the entrance to the cell, R_(σ)(t)—change of the admixture concentration in the flow of the fluid being studied inside the measurement cell, K(t)—the impulse response of the measurement cell.

The relation of a physical property of the fluid being studied to a concentration of the admixture may be determined preliminarily; in this case, the change of the admixture concentration in the fluid flow inside the measurement cell is determined by measuring the physical property of the fluid being studied.

The measured fluid property might be an electrical conductivity, a density, a radioactivity, etc.

A quality of the measurement cell may also be estimated; for this purpose, the difference between a concentration measured inside the measurement cell and a concentration at the entrance to the measurement cell is found. The obtained value indicates the quality of the measurement cell.

The quality of the measurement cell may also be estimated by determining the impulse response of the measurement cell applying the Fourier transformation technique and comparing the Fourier transformation from the function K(t) with the constant 1/√{square root over (2π)}.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention is explained by drawings:

FIG. 1 shows a sample layout of a test unit with a measurement cell;

FIG. 2 shows the results of measuring the admixture concentration in the fluid for two measurement cells;

FIG. 3 shows the results of measuring the admixture concentration in the fluid in the first cell and the results of determining the concentration at the entrance to the measurement cell applying the suggested technique;

FIG. 4 shows the results of measuring the admixture concentration in the fluid in the second measurement cell and the results of determining the true concentration applying the suggested method; and

FIG. 5 shows the results of determining the true concentration in the two cells applying the suggested method.

DETAILED DESCRIPTION

An example of applying the invention in measuring a fluid electrical conductivity is provided. In the example, a measurement cell consisting of a dielectric material tube and two electrodes being in contact with the fluid is used (FIG. 1). Such a cell design is used in a number of measurements (see, for example, Zhiyao Huang, Jun Long, Wenbo Xu, Haifeng Ji, Baoliang Wang, Haiqing Li, Design of Capacitively Coupled Contactless Conductivity Detection Sensor//2012, Flow Measurement and Instrumentation, in press). The fluid electrical resistivity inside the tube is associated with the concentration of the admixture in the flow. So, the resistivity measurements allow determining the admixture concentration at the volume limited by the tube and the electrodes. The admixture concentration at the entrance to the cell (the true concentration) may differ considerably from the readings for the cell measuring the volume of the concentration due to the heterogeneity of the concentration field. When using such a cell, the difference between the cell readings and the true admixture concentration at the entrance to the cell should be reduced.

The measurement cell is a signal processing system. Presumably, the processing system possesses a linearity property and does not depend on time (a Linear Time Independent System, hereinafter LTIS; for examples of such systems see J. P. Hespanha, Linear Systems Theory//2009, Princeton University Press, 263 p., ISBN 978-0-691-14021-6). If the LTIS impulse transient function is known, then for any measured output signal of LTIS (a response to an input signal) the corresponding input signal might be restored with the deconvolution method. In its turn, the impulse transient function of LTIS may also be calculated with the deconvolution method measuring the LTIS response to an input signal known in advance.

The input signal of this LTIS is the admixture concentration in the carrying fluid at the entrance to the cell; the output signal is the admixture concentration calculated based on the measured resistance of the carrying fluid. The LTIS is peculiar of an impulse transient function often called an impulse response of the system.

The invention allows determining a change of the admixture concentration in a carrying fluid at the entrance to a measurement cell. The measurement procedure involves a number of stages. A relation of an electrical conductivity of the fluid being studied to the admixture concentration is determined preliminarily. A fluid, for which the change of the admixture concentration in time at entering the cell is known, is pumped through the cell, and the system output signal is registered (i.e., of the measurement cell). Then, the system impulse transient function is determined commonly applying the deconvolution method. The system output signal is then registered (i.e., the resistance) for the fluid being studied with an unknown change of the admixture concentration at the entrance. Finally, the unknown change of the admixture concentration at the entrance to the cell is found applying the deconvolution method and using the impulse transient function found before, and the concentration in the cell found during the resistivity measurements.

To apply the invention, a test unit for reproduction of the flow with the admixture shown in FIG. 1 was used. The unit consists of a pump system 1 for the fluid with the known change of the concentration at the entrance to the measurement cell; a pump system 2 for the fluid being studied, the concentration of which needs to be determined; a three-way crane 3; a porous medium 4; a measurement cell 5; an ohmmeter 6; electrodes 7; a dielectric material tube 8 and a backpressure system 9.

The admixture concentration field is time-dependent. Determination of a change of concentration in a particular flow point allows estimating the admixture dispersion inside a porous sample. The admixture concentration may be determined by the change of the fluid electrical resistivity inside the measurement cell. The concentration determination error is expressed in σ.

An empirical relation of the resistance to the admixture concentration is found in advance (for example, to the fluid salinity). The relation is used in estimating the admixture concentration by, for example, the ohmmeter readings in determining the concentration of the electrically conductive admixture (FIG. 1).

The flow, for which the dynamics of the admixture distribution is known and the change of the admixture concentration at the entrance to the measurement cell (equal to i(t), where t is time) is known, is then run on the unit. The cell registers the resistivity change value r_(σ)(t). So, both functions i(t) and r_(σ)(t) are known.

The performed experiment data are used for recovering the impulse response of the system K(t) (the impulse transient function of LTIS):

∫_(o) ^(t) K(t−τ)i(τ)dτ=r _(σ)(t)   (1)

∥r−r _(σ)∥<σ

If i(t)≈1 in a known flow regime, the core is easily found based on the result of measurements as

${K(t)} \approx {\frac{r}{t}{(t).}}$

Otherwise, the deconvolution method should be used to recover the core K(t), i.e., solving the integral equation assuming smoothness of the required function.

An experiment is then performed, in which the distribution of the admixture in the flow should be considered. The change of the admixture concentration at the entrance to the measurement cell is I(t); the change of the concentration registered during the measurements is R(t). The function I(t) is unknown.

The input signal I(t) is to be found with the known transient impulse function K(t) and the output signal R(t) in the form of a measured concentration (see equation (2) below). The task is solved with the function I(t), which estimates the local change of the admixture concentration in the flow.

∫_(o) ^(t) K(t−τ)I(τ)dτ=R _(σ)(t)   (2)

The difference between the input and the output signals may serve as an estimate of the measurement cell quality. If the set of measurements {I^(n)(t), R^(n)(t)} is available, the difference R^(n)(t)−I^(n)(t) for the flows under study may be calculated (n is the experiment number). This allows estimating the measurement cell quality. The lower is the difference, the higher—the measurement cell quality.

The measurement cell quality may also be estimated after recovering the impulse response. The highest measurements quality corresponds to the case when R_(σ)(t)=I(t). Using the equation (2), we get K(t)=δ(t), where δ(t) is the Dirac function. It is known that the Fourier transformation for the Dirac function is F[δ(t)]=1/√{square root over (2π)}. Thus, the closer the Fourier transformation from the function K(t) to the constant 1/√{square root over (2π)}, the higher the measurements quality. The distance between the functions may be estimated, for instance, by the L²-norm:

${{f_{1} - f_{2}}} = \left( {\int_{0}^{t}{\left( {{f_{1}(\tau)} - {f_{2}(\tau)}}\  \right)^{2}{\tau}}} \right)^{\frac{1}{2}}$

The suggested procedure was applied at flooding core with a saline solution of various concentrations. The 40 g/l NaCl solution was used as base fluid; the admixture was modeled with 60 g/l NaCl solution. A core sample (a porous rock sample) was put in a sealed core holder. The measurement cell was positioned successively behind the core holder in the flow pattern and registered the admixture concentration in the flow behind the core holder.

The first measurement cell (cell 1) consisted of a plastic tube with steel electrodes on the ends (FIG. 1). The known flow regime was a distribution of the admixture in the measurement cell with no core provided in the flow pattern. The concentration at the entrance to the measurement cell changed abruptly from 0 to 1, so the input signal is the Heaviside function. The signal registered in the measurement cell is the result of smoothing the input signal. The first cell has the relaxation time, within which the readings changed from 0 to 1 at an abrupt change of the concentration at the entrance. The time scale was the time of pumping a single porous core volume; the time scale was 5 s at 2 cm³/min pumping rate (FIG. 2, cell 1).

The use of the porous material (core) in the experiment results in washing the front out with smooth increase of the concentration at the outlet from the core holder. The concentration at the entrance to the cell was estimated applying the suggested method (FIG. 3, cell 1). It differed from the measured concentration considerably (the absolute error between the actual concentration at the entrance to the cell and the measured concentration was 10%).

The suggested technique was checked by comparing the results for two different measurement cells. The core flooding experiment was repeated using an improved measurement cell, in which the plastic tube was replaced with a glass tube of a smaller volume (FIG. 4, FIG. 5, cell 2). The improved cell had a lower relaxation time compared to the first cell (FIG. 2, cell 2). The technique was applied to the concentrations registered with two different cells for the purpose of estimating the admixture concentration at the entrance to the measurement cell. In the course of the experiment it was found that the recovered true admixture concentrations at the entrance were less than 3% different. The study performed with the application of the invention showed that the improved cell increased the quality of measurements and provided the precision required for the core flooding experiment.

The difference between the input and the output signals may serve as an estimate of the measurement cell quality. The input signal is recovered with the suggested technique applying the deconvolution method. The difference between the signals may be calculated with various functional norms, for example, the norm L² equal to the integral of the square of the differences of signals by a time interval. The minimum value of difference corresponds to the most precise measurement. Thus, when processing a series of experiment data applying the deconvolution method we may estimate the quality of the measurement cell. 

1. A method for determining local changes of admixture concentration in a fluid flow, comprising: pumping a fluid containing an admixture through a measurement cell, a change of concentration of the admixture in time for this fluid at an entrance to the measurement cell being known, determining a change of concentration of the admixture in time inside the measurement cell, recovering an impulse response of the measurement cell applying the deconvolution method, pumping a fluid being studied through the measurement cell and determining a change of concentration of the admixture in time in a flow of the fluid being studied inside the measurement cell, and determining a change of concentration of the admixture in time in the flow of the fluid being studied at the entrance to the measurement cell with the following equation: ∫₀^(t)K(t − τ)I(τ) τ = R_(σ)(t) where τ—an integration variable, t—time, I(t)—change of concentration of the admixture in the fluid flow at the entrance to the cell, R_(σ)(t)—change of concentration of the admixture in the fluid flow inside the measurement cell, K(t)—the impulse response of the measurement cell.
 2. The method of claim 1, wherein preliminary a relation of a physical property of the fluid to the admixture concentration is determined, and the change of the admixture concentration in the flow of the fluid being studied inside the measurement cell is determined by measuring the physical property of the fluid being studied.
 3. The method of claim 2, wherein the physical property being measured is an electrical conductivity of the fluid being studied.
 4. The method of claim 2, wherein the physical property being measured is a density of the fluid being studied.
 5. The method of claim 2, wherein the physical property being measured is a radioactivity of the fluid being studied.
 6. The method of claim 2, wherein additionally a quality of the measurement cell is estimated.
 7. The method of claim 6, wherein a difference between the admixture concentration in the fluid being studied measured inside the measurement cell and the admixture concentration in the fluid being studied at the entrance to the cell is determined and using the obtained value for estimation of the quality of the measurement cell.
 8. The method of claim 6, wherein the impulse response of the measurement cell is determined by Fourier transformation and comparing the Fourier transformation from the function K(t) is compared with the constant 1/√{square root over (2π)}. 